What is the Newton-Raphson Method?

The Newton-Raphson Method (or simply Newton's Method) is a way to approximate the zeroes of the function. We can use it as an alternative to solve functions or equations.

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In this post, we'll explore a technique to find the zeroes of a function: the Newton-Raphson Method (or simply Newton's Method).

Deriving the Formula

Let's say we are to find the zero of a function \(f\) using this method:

Assume any value \(x_1\). It doesn't matter how big or small it is.
Consider the point on the function corresponding to \(x_1\) and draw a tangent line. Formulate its linear equation using the point-slope form: \(m=\frac{y-y_1}{x-x_1}\)
Let \(m\) be the function's first derivative, \(f\) and \(y_2\) be the value of \(f(x_1)\): \(f^{\prime}(x)=\frac{y-f(x_1)}{x-x_1}\)
Rearrange the equation so that we isolate \(x\). In addition, we let \(y=0\). Our goal is to compute the value of \(x\). We want to find the value of \(x\) when the tangent line crosses the x-axis: \(x=x_1-\frac{f\left(x_1\right)}{f^{\prime}\left(x_1\right)}\)
The computed \(x\) is our first estimate of the zero of the function. Let's call this \(x_2\).
Improve the estimate by repeating steps 2 to 5 as often as required. Returning to step 2, consider the point on the graph with \(x_2\).

If you continue to repeat this process \((x_3, x_4, x_5, x_6, …, x_n)\), you'll notice that the computed \(x\) values get closer and closer to the zero of the function.

We can summarise this method with the general formula:

\(x_{n+1}=x_n-\frac{f\left(x_n\right)}{f^{\prime}\left(x_n\right)}\)

\(x_{n+1}\) is the x-value of the next coordinate
\(x_n\) is the x-value of the original coordinate
\(f\left(x_n\right)\) is the evaluated function with \(x_n\)
\(f^{\prime}\left(x_n\right)\) is the evaluated first derivative of the function with \(x_n\)

The more repetitions we make with the formula, the more accurate the result is. It will eventually converge to a singular x-point, the function's zero.

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Newton's Method May Fail

This method may fail under two conditions:

The function's first derivative evaluated at the assumed value is zero, \(f^{\prime}(x_1)=0\). If we look at the general expression, the denominator \(f^{\prime}(x)\) cannot be zero (division by zero is impossible).
After performing multiple iterations, the computed values diverge from each other (doesn't meet on a singular point).

Example

Say we want to find the zeroes of \(x^2+2x-4=0\). From the degree of the polynomial function, there would be two zeroes.

To start, we need to assume a zero value of the function. We begin with \(x=2\) and perform the above steps. Eventually, it will converge to a zero equal to 1.236...

\(x_{n+1}=x_{n}-\frac{f(x_{n})}{f^\prime(x_{n})}\)

\(x_{n+1}=x_{n}-\frac{x_{n}^2+2x_{n}-4}{2x_{n}+2}\)

First Iteration

\(x_2=2-\frac{(2)^2+2(2)-4}{2(2)+2}\)
\(x_2=1.333...\)

Second Iteration

\(x_2=1.33-\frac{(1.33)^2+2(1.33)-4}{2(1.33)+2}\)
\(x_2=1.238...\)

Third Iteration

\(x_2=1.238-\frac{(1.238)^2+2(1.238)-4}{2(1.238)+2}\)
\(x_2=1.236...\)

To find the other zero, we assume another value. Say, we start with \(x=-2\). Performing the steps above, we will have a zero of -3.236...

Applications

Now that we know the general procedure, where do we see this method? A great example would be scientific calculators. Most of them solve for the zeroes of the function using this way.

Input a function expression in the calculator.
When we try to solve it, the calculator will ask for an \(x\) value first. It is the starting point of the calculation.
The calculator will perform the necessary procedure behind the scenes.
Afterward, it will give the zero of the function.

Summary

The Newton-Raphson Method (or simply Newton's Method) is a way to approximate the zeroes of the function. We can use it as an alternative to solve functions or equations.

We can summarise this method with the general formula: \(x_{n+1}=x_n-\frac{f\left(x_n\right)}{f^{\prime}\left(x_n\right)}\). \(x_{n+1}\) is the x-value of the next coordinate, \(x_n\) is the x-value of the original coordinate, \(f\left(x_n\right)\) is the evaluated function with \(x_n\), and \(f^{\prime}\left(x_n\right)\) is the evaluated first derivative of the function with \(x_n\)

We first assume an \(x_n\) value and repeatedly use the formula to arrive at zero.

The more repetitions we make with the formula, the more accurate the result is. It will eventually converge to a singular x-point, the function's zero.

This method may fail under two conditions: (1) the function's first derivative evaluated at the assumed value is zero, \(f^{\prime}(x_1)=0\), and (2) after performing multiple iterations, the computed values diverge from each other.

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